Time lapse recordings of cultures at different ages revealed massive convergence of cell processes, either by the growth of processes towards preexisting contact sites between other processes (Figure 1a) or by the lateral movement of several processes towards a single area (Figure 1b). Such behavior resulted in the formation of clusters, several microns in width, comprising contact sites of multiple processes that remained stable over the course of several weeks (Figure 1c), a period corresponding to almost the entire life span of the cultures. In the cultures, cluster sites appeared sufficiently frequently so as to cause a ‘patchy’ network morphology (Figures 1d, 1e). A second type of process behavior involved the fasciculation of two or more processes over long distances, creating stable bundles (Figure 1b).

Efforts next focused on defining the composition of the converging processes, using specific antibodies directed against dendritic (microtubule–associated protein 2 (MAP2)) and axonal markers (Neurofilament (M NFM)). The clusters were thus shown to be comprised of numerous converging dendrites originating from many cells (Figure 1f, 1g), together with axonal (Figure 1h) bundles, but devoid of cell bodies. These convergence sites, termed dendro-dendritic contact clusters (DCCs), were first characterized in terms of dendritic behavior.

In the first week of culture, DCCs contained few dendro-dendritic contacts (DCs), formed by a small number of single dendritic branches (Figures 2a, 2b). Over the next weeks, massive DCCs appeared, containing dozens to hundreds of DCs formed by convergence of dozens of single and bundles of dendritic branches into areas a few tens of microns in width (Figures 2c, 2d). Dendritic behavior was thus assessed by measuring the rate of DC aggregation. Based on the methodology described in Shefi et al. [17], images of dendritic networks were converted into graphs by defining DCs as nodes and the dendritic segments connecting them as edges (Figure 2e). The relative neighborhood density index, Ω_x, an estimator of aggregation (or dispersion) [18], [19], was calculated by measuring the density of DCs found in concentric rings of defined radius (x) around each DC, then averaging over all DCs in the network. Comparing between networks was possible after normalizing by each networks average DC density. This calculation yields the level of ‘network aggregation’ (see Methods and legend to figure 2f for further explanation). As shown in the inset to Figure 2f, network aggregation levels increased with culture maturation. The average aggregation within 0–5 µm from a given DC reached 1.57±0.1 after 5 days in vitro (DIV), whereas after 12 and 21 DIV, this value increased to 2.01±0.1 (mean±95%, C.I.) and 2.74±0.29, respectively. The increase in network aggregation during culture development was maintained over the entire analyzed range of ring radii (0–30 µm, Figure 2f). Hence, during growth in culture, dendrites tend to accelerate their rates of convergence and contact with other dendrites.

Ω_0–5 values were computed in 12 DIV cultures treated with different inhibitors of synaptic activity and found to be 25% (TTX), 29% (CNQX), and 22% (NMDA) below control, with the overall number of DCs being maintained (p>0.05, Figure 2h). Thus, DC aggregation is activity promoted.

The challenge in quantifying DCCs lays in defining their boundaries and deciding which DCs belong to a given cluster. Such decisions were made through ‘Hierarchical cluster analysis’, a widely used data-partitioning tool [20], [21] and subsequently confirmed by the ‘Silhouette’ metric approach [22], see also Supplemental Material S1 section 3. These criteria enabled us to discern between significant and non-significant data partitioning, assigning values of 1 to perfectly clustered data, 0 to ambiguous results and −1 to poorly partitioned data (Figure 3a1, a2, b1). For the sake of robustness and conservativeness, we opted to consider for further analysis only those DCCs with silhouette values greater than or equal to 0.6. This cut-off value, accepted for various types of biological data analysis [20], [23], [24], provided a DCC selection which best fitted our visual selection performed. Since DCC analysis depends on both the minimal number of DCs to be considered as a DCC and the maximal distance allowable for including a DC into a DCC, DCCs were subdivided as small, medium and large (Figures 3b,c), based on both threshold parameters (see Methods). Using these criteria, we compared DCC probability of appearance in 12 DIV cultures to that obtained in a computer simulation of randomly-distributed dendritic trees presenting a similar DC density as in the cultures (see Supplemental Material S1 section 1). Our comparison showed that the abundance of the three DCC classes (normalized per total dendritic length) in culture was higher than in the simulations, as shown in Figure 3d.

Next, the effects of inhibiting synaptic activity on: i) DCC density – the number of identified DCCs normalized to total dendritic length in each field and ii) DC clustering - the proportion of DCs that participate in DCCs, as a fraction of the total number of DCs in a given network, were considered. In general, DCC density and DC clustering decreased in response to all the activity inhibitors tested, with variations among the three DCC groups being observed (Figures 3e, 3f). APV was the strongest inhibitor of DCC density, being the only one to reduce that of all groups (Figure 3e). DC clustering showed more homogeneous sensitivity to the various antagonists used (Figure 3f). Thus, DCC formation is an activity-enhanced process.

To check whether dendritic convergence influences synaptic distribution, dendrites were co-labeled with antibodies to synaptophysin (Sph, a pre-synaptic marker) and to the GluR2 and NR1 subunits of AMPA and NMDA glutamate receptors and to post-synaptic density 95 protein (PSD-95) (post-synaptic markers). Antibody labeling clearly showed accumulation of pre- and postsynaptic proteins in DCCs to a higher level than found along dendritic segments located beyond the DCCs (Figure 4a–c). Synaptic clustering in the same type of culture was previously reported [25] but was not related to the manner of dendritic tree interaction. Correlation between DC aggregation and synaptic distribution was next assessed. The synaptic level at a DC was considered as the total fluorescence measured in a disk 5 µm in radius, centered at the DC. As reflected in Figure 4d, positive correlation between these two parameters was observed for both pre- and post-synaptic markers (0.25±0.03, 0.34±0.05 and 0.20±0.06, respectively). This correlation was diminished or completely abolished by treatment with CNQX or APV (Figure 4e). Thus, aggregation of DCs in the cultured network is associated with activity-enhanced synaptic clustering.

We next considered whether such clustering reflected an increase in synaptic density specifically at the DCCs. To measure that, DC-associated synaptic distribution at DCCs was computed as described in supplemental material S1 section 2. Briefly, the procedure involved the following steps: 1. Building a binary mask around the area covered by the dendritic network; 2. Computing the ‘In cluster’ mean intensity by defining DCC, as described above, creating a polygon with vertices around the most extreme DCs, summing the fluorescence intensity within the polygon and dividing it by the DCC's area. The average DCC intensity was then divided by the number of DCs found in the polygon. To keep track of the portions of the dendritic mask not included in DCCs, the polygon was then removed from the binary mask (computed in step 1); 3. Computing the ‘Out of cluster’ average was done by summing the total fluorescence in the remaining of the binary mask and dividing it by its area, and finally by the number of DCs which were not associated with any DCCs. The results showed Sph, NR1 and GluR2 to be consistently higher inside than outside a DCC, albeit to varying extents among the different DCC categories (Figure 4f). Hence, synaptic density is enhanced at DCCs compared to other areas along the dendrites.

The question whether synapses clustered at DCC were of different strengths than other synapses was addressed by measuring presynaptic secretion levels using the synaptic vesicle recycling indicator, FM1-43. Distances between dendrites were tightly related to the distribution of FM1-43-positive puncta (Figures 5a–c), with terminals showing the highest endocytotic (Figures 5d, e) and exocytotic (Figures 5f, g) capacities being located at DCCs. Moreover, the size of individual terminals was higher at than out of DCCs (Figure 5h). We dissected out hippocampal sub-regions and measured secretion level from presynaptic sites within and outside DCCs in cultures of the mossy fiber and the Schaffer collateral pathways. Secretion level was measured by subtracting FM1-43 total fluorescence intensity in terminals after induction of release by K⁺ from that imaged prior to release (see full explanation in the method section). We found that terminals within DCCs had 2.8- (mossy fiber) and 2.2- (Schaffer Collateral) fold higher secretion level than terminals located outside DCCs, respectively (Figure 5i). These results indicate that synaptic connections are stronger at than outside DCCs.

We wondered if the high dendritic proximity at the center of DCCs could ease the way for axons to switch targets. Looking first at DCs, we found that axons fasciculating with a crossing dendrite often defasciculated near the DC site, turned and fasciculated with the other dendrite (Figures 6a, 6b). Similarly, axons reaching small DCCs through fasciculation with one of the converging dendrites often left the DCC to fasciculate on another dendritic branch, while at the DCC core, additional dendritic branches were contacted (Figure 6c). At large DCCs, axons form complex meshes (Figures 6d, e), within which the axons frequently turned and changed direction (Figure 6e). This behavior, combined with the high dendritic proximity at the center of the DCC, led to a situation where single axons switched targets and contacted many different dendritic branches in a relatively short length (Figure 6f). We noted that the density of intersections axons make with dendrites at DCCs is on average 2.5-fold higher than outside the DCCs (Figure 6g). Moreover, being close to several dendritic branches, many individual axonal terminals at DCCs made multiple synapses on different branches (Figure 6h). Hence, the presence of DCCs increases the number of neurons innervated by single axons in the cultured network.

To check the role of dendritic convergence in shaping network architecture, we adopted the ‘Economical Small World Network (ESWN)’ analysis [26], [27]. This method estimates the overall efficiency of connectivity scheme in a network by providing information on its potential to propagate signals and its robustness (for a detailed explanation, see Supplemental Material S1 section 4a): 1. global efficiency – a value between 0 and 1 which reflects the speed of parallel information spread across entire networks and is inversely proportional to the average length of ‘shortcuts’ among distant network nodes. A value of 1 is assigned to an ideal (all-to-all) network, where all nodes are interconnected, and the length of each ‘shortcut’ between any two DCs is the shortest one. Departure from the all-to-all configuration yields lower values (closer to 0); 2. local efficiency – a value between 0 and 1 which is proportional to the number of local neighbors to which a given DC is connected [28]; and 3. wiring cost - a surrogate of the investment required for building a network of specific local and global efficiencies. In the case considered here, this value would correspond to a combination of dendritic length and the synaptic clustering (see Supplemental Material S1 section 4b). For a network to be considered as an efficient or economic SWN, its global and local efficiencies should lie in the proximity of 0.3 and 0.7, respectively, attained under a wiring cost below 0.3 [26], [27]. We found that all analyzed dendritic networks acquired an ESWN configuration (Figure 7a) while attaining higher efficiencies and lower wiring costs than other neural networks, such as the entire C. elegans somatosensory network or the cortico-cortical connections of both the macaque and cat [26], [27]. The efficiency values did not scale with network size (i.e. the number of DCs), whereas the cost value decreased with an increase in network size, pointing to a conserved organizational pattern (Figure 7a). In addition, the ESWN topological property of the cultured networks was proven to be activity-independent, as pharmacological inhibition of synaptic activity did not affect the measured values, relative to control conditions (not shown). Thus, the results show that dendritic trees in culture overlap to form a geometrical architecture that yields a high level of connectivity and ‘shortcuts’ among their DCs.

To assess whether the clustering of synaptic connections at DCCs is related to the network ESWNs configuration, two numerical experiments were performed. First, the efficiency analysis was redone after disrupting the clustering by randomly shuffling the synaptic weight assigned to each vertex. The reshuffling process was repeated 30 times for each network and an average parameter value was reported. Results from this numerical experiment design were best analyzed in a paired test, where it was possible to distinguish between pre- and post-shuffling effects on each individual trial. In all cases, disrupting the spatio-synaptic configuration always led to a decrease in global efficiency (6.09%±0.23; Figure 7b). Similarly, shuffling always led to a small, though significant, decrease in the local efficiency (0.4%±0.16). Wiring cost in the shuffled configurations was 0.95%±.1.73 lower. Thus, the results imply that dendritic networks self-organized into an efficient configuration by means of matching between the geometrical distribution of their contacts and that of synaptic sites.

In a complementary approach, an artificial scenario was created to demonstrate that matching between dendritic and synaptic distributions could emerge as a wiring principle for the optimization of network organization. A genetic algorithm (see Supplemental Material S1 section 5) was implemented to determine that configuration of vertex weights that simultaneously maximized both E_glob and E_loc of a lattice-like network under two different spatial configurations (i.e. regular and aggregated; Figures 7c1 and c3, respectively). The aggregated configuration presents the same topology as the regular case, only with the formation of a cluster of aggregated vertices at its center (Figure 7c3). The optimal configuration for the aggregated case (Figure 7c4), but not for the regular (Figure 7c2), averaged over 50 trials, resulted in a clustering of the highest vertex weights assigned to the vertices at the lattice core. Two additional clusters of vertices with high weights were formed at the lattice ridge, at a location where high weight vertices were also present in the regular case. The distribution of vertex weights did not differ between the two cases and showed bimodal-like behavior (Figure 7d1). The same relationship was, furthermore, observed between each vertex weight and the mean weight of its subgraph (i.e. the set of adjacent neighbors; Figure 7d2). Combined, these results demonstrate that matching between the distribution of dendrites and synapses in the culture contributes to the efficiency of the dendritic network SWN configuration.

We then checked if DCCs exist in vivo. A first step considered whether DCCs are not merely a product of cell growth on a 2D surface. Confocal microscopy was employed to seek DCCs in adult rat brain. Due to their high dendritic density, areas like the hippocampus and the cortex were difficult to analyze, so we instead chose to focus on more disperse regions, like the Substantia Nigra. Because convergence in the brain occurs in three dimensions, our first approach was to collect sections from a 50 µm×50 µm area at a depth of 9 µm and merge them into a single image. Taking the average width of thin dendritic branches as 1 µm, the average distance between three converging dendrites at a 9 µm depth is about 3 µm, encompassing two synaptic connections, each arriving from a different dendritic branch. Using such values for dendro-dendritic contacts in 3D, we detected areas within the substantia nigra containing up to 8 convergence sites (equals to 3.2×10⁵ per mm³), each including 3–6 dendritic branches (Figures. 8a, 8b). On average, 33% of the dendritic segments located within the analysis area were involved in at least one convergence site and 12% were involved in more than one (Figure 8b, red arrows). DCC density in simulations of random 3D distribution of cylinders (with the same dimensions and density as in Figure 8a) was 15-times lower (not shown). We also found sites in the Substantia Nigra containing higher numbers of converging dendrites, associated with clusters of puncta expressing synaptophysin (Figures. 8c, 8d) at a higher level than those found beyond the convergence site (Figure 8e).

As DCCs were originally found in culture, we thought it logical to search for DCCs in more primitive neuronal systems than the brain, such as ganglions of the lobster stomatogastric system. Here, we found 6 convergence sites (3–4 processes each, Figure 8f) per 100 µm³ cube. Both the size and serotonin levels of synaptic puncta (Figures 8f, 8g) were higher at the convergence site that elsewhere.

Together, these results suggest the existence of convergence behavior in vivo, similar to that found in culture.

Studies describing dendritic morphology based on analysis of single dendritic trees often have led to the conclusion that dendritic ramification is random and that the growth directionality is unbiased toward specific targets [32], [33]. We present here a different explanation for dendritic tree morphogenesis, where the interactions of a tree with other trees are major players in the design of the final dendritic morphology. According to our model, the growth of dendritic branches is preferentially directed toward areas of high dendritic proximity to form bundles and DCCs. Thus, the development of particular dendritic tree architectures can be predicted by considering the distribution and density of DCs around the growing trees. By the same token, the morphology of entire networks of dendritic trees can be described by considering the number, location and size of their DCs, bundles and DCCs. Thus, studying dendritic proximity maps may enable us to proceed beyond the structure of individual dendritic arbors to that of full dendritic networks.

Interestingly, we did not detect DCCs that were formed by convergence among sister branches of a single dendritic tree. Rather, DCCs always included non-sister branches of several neighboring dendritic trees (Figures 1f and 2a). This ‘non-self’ manner by which dendrites associate indicates that sister branches undergo ‘self avoidance’, and that by preferentially converging with non-sister branches to form DCCs they highly increase the overlap among different dendritic trees.

In relating to dendritic proximity by describing dendritic networks as graph of connections among DCs (Figure 2), we were able to show that in culture, such networks assemble into ESWN configurations (Figure 7). The main anatomical consideration of such a configuration is that a dendritic network exhibits ‘shortcuts’ that connect distant dendrites. Such an arrangement would have significant implications for axonal directionality and patterning, as many of the axons fasciculate with dendrites and follow their tracks (Figure 6). This means that if axons have access to ‘shortcuts’, their chances of innervating distant dendrites are increased, enhancing the connectivity of the entire network. This possibility also fulfills the ‘minimal wiring length’ principle guiding the wiring of many neuronal networks in the central nervous system [34]–[36].

However, the major contribution of the structural organization of dendritic networks to connectivity arises from the convergence of dendrites into DCCs. As reflected in the example presented in Figure 6f, axons reaching DCCs frequently contact many (up to several dozens, depending on the DCC size) of the converging dendrites. Due to the high proximity of dendrites, only a few microns of axonal growth and turning suffice for the axons to switch targets. Such an increase in target/axon ratio can be attributed to their minimal wiring length property, but also means that single cells would connect to a higher number of neurons in the network than would be the case in non-aggregated networks. The outcome of this wiring mechanism may be an all-to-all connectivity.

The convergence of dendrites, as well as its influence on synaptic distribution, may be of physiological relevance for several reasons: 1. As mentioned in the Introduction, the high dendritic proximity found at DCCs likely facilitates ephaptic coupling among the converging dendrites. 2. Clustering of synaptic connections at DCCs results in local reduction of inter-synaptic distances, as compared to those existing outside DCCs. It has previously been shown that clustered synapses exhibit firing properties of lower amplitude and higher frequency [37], as well as higher summation capability [38], as compared to non-clustered synapses. In this scenario, the high frequency of DCCs, especially in mature cultures, may play a significant role in activity patterning. 3. It is possible that at DCCs, due to the high density of synaptic connections and dendrites, glutamate spillover [39], [40] becomes an effective mechanism for activating multiple dendrites. 4. DCCs contribute to the pattern and total level of synaptic strength in the network, as synaptic connections at DCCs are richer in secretion capability (Figure 5) and in pre- and postsynaptic markers (Figure 4), as compared to those in non-converging dendrites.

In addition, taking into account that DCC formation is promoted by synaptic activity (Figure 3), these localized synaptic markers enrichments suggest that plasticity-related synaptic strengthening events transpire at these sites. This possibility is further supported by the fact that the formation of DCCs is partially mediated by synaptic activity through the NMDA receptor (Figure 3e), but the accompanied synaptic clustering and strengthening at the DCCs is promoted by AMPA receptors (Figure 3f). Such a distinction in glutamate receptors function is similar to that found in long-term potentiation of the mossy fiber pathway, where APMA receptors, and not NMDA underlie the plasticity induction [41], [42].

Combined, the above possibilities suggest that the map of dendritic proximities plays a role in patterning the activity of neuronal networks, as well as their plasticity. This hypothesis is supported by the findings that matching the geometrical architecture of dendritic networks and the corresponding synaptic distribution yields an ESWN configuration for the dendritic network (Figure 7a), a property which could enhance activity synchronization [43]–[46]. In addition, as a result of such matching, ESWN dendritic networks offer ‘shortcuts’ between distant parts of the network presenting the highest synaptic weight/length ratio (Figure 7). This idea is strengthened by the observation that when global and local efficiencies are maximized in a ESWN model, clustered distribution of synaptic connections is obtained (Figure 7c). A probable outcome of the existence of such ‘shortcuts’ is that dendrites following these shortened pathways should have a higher chance of interacting with synaptic clusters and, thereby, increase the density of inputs than had they chosen a ‘non-shortcut’ pathway. Moreover, as connections at DCCs are of higher strength than elsewhere, ‘shortcut’ pathways may increase the chance of dendrites to be synaptically activated. Hence, dendritic convergence promotes synaptic weight-based ESWN organization, thereby patterning and increasing the strength of connectivity and activity in neuronal networks.

We conclude that the proximity among dendritic branches of neighboring neurons is a functional structural entity. Being up-regulated by synaptic activity and associated with enrichment in synaptic density and strength, dendritic proximity affects the conversion of synaptic information into a map of synaptic connections and synaptic strength distributions. Accordingly, when neuronal network activity increases, the network architecture becomes more aggregated through DCC and bundle formation, leading to an increase in synaptic clustering and strength. This structure-mediated, activity-dependent synaptic strengthening may serve as a novel structural-based mechanism of plasticity.

CA3 and dentate gyrus from brains of postnatal rats were treated with trypsin (Sigma, type XI), triturated, and the released cells were plated (2×10⁵ cells/ml for 2D and 3×10⁸/ml for 3D cultures) onto glass cover slips or coral skeletons coated with poly-D-lysine (Sigma, 20 µg/ml) and laminin (Collaborative Research, 10 µg/ml), in a culture medium containing 10% serum, as described previously [47]. Cultures were maintained in a 37°C incubator in a 5% CO₂ atmosphere in the presence or absence of CNQX (10 µM), APV (50 µM) or TTX (1 µM).

For immunocytochemical study, cells were labeled as described previously [48]. Briefly, cells were fixed for 10 min at room temperature with 4% paraformaldehyde, permeabilized with 0.25% Triton X-100, and blocked with 3% normal goat serum. The cells were then incubated overnight at 4°C with anti-MAP2 (1 µg/ml) (monoclonal, Sigma, Oakville, Ontario, Canada); anti-NFM, anti-GluR2, anti-NR1, anti-PSD-95 (0.5 µg/ml) (polyclonal, Chemicon, Temecula, CA, USA) or anti-synaptophysin (0.5 µg/ml) (polyclonal, DAKO, Mississauga, Ontario, Canada) antibodies. Immunolabeling was visualized with secondary antibodies conjugated to Alexa-488 or Cy3 (2 µg/ml) (Molecular Probes, Eugene, OR, USA).

Images were obtained using Axiovert 200 M microscopes with Plan-Neofluar 20×/0.5 and Plan-Apochrome 63×/1.4 objectives, equipped with 12-MHz CCD cameras (SensiCam, PCO, Kolheim, Germany). Acquisition and analysis were performed under Metamorph v6.3 (Molecular Devices, USA) or via custom-made MATLAB 7.0 software (Matworks, Massachusetts MA, USA). Figures were processed using PhotoShop 7.0 (Adobe Systems, San Jose, USA).

Cultures maintained for 14–16 DIV were exposed for 30 sec to 15 µM of the synaptic vesicle-recycling marker, FM1-43 (Molecular Probes, Carlsbad, USA), in HEPES/Tyrode's buffer and 90 mM K⁺. After a 3 min wash, ‘uptake’ images from two to four randomly selected fields were acquired under non-saturating pixel value conditions. FM1-43 was then secreted by application of 90 mM K⁺, followed by a 3 min wash with Tyrode's buffer and the previously acquired fields were imaged again under the exactly the same imaging conditions (i. e. illumination setting, exposure time and binning); this defined the ‘release’ set of images. Finally, vesicle-recycling activity was estimated by computing the ‘net release’ image, obtained by subtracting the ‘release’ from the ‘uptake’ images.

Cultures were transfected between 5–10 DIV, as described previously [49]. In brief, cells were washed and incubated for 45 min at 37°C with MEM containing 0.5% glucose. Each coverslip was then incubated for 30–40 min at 37°C with 80 µl of DNA solution until a heavy precipitate formed (DNA solution: 5 µg DNA (plasmid pIRES2-EGFP, Clontech), 250 µl of 250 mM CaCl₂ and 250 µl of BBS (in mM: NaCl 280, Na₂HPO₄ 1.5, BES 50, pH 7.1)). Finally, cells were washed twice with HBS and twice with MEM, and then returned to their original growth medium. Cells were imaged 5–8 days after transfection.

Primary fixation was done with modified Karnovsky's fixative (2.5% glutaraldehyde, 2% paraformaldehyde in 0.1 M cacodylate buffer (pH 7.2) for 30 min at 37°C. The samples were post-fixed with 1% osmium tetroxide at 4°C for 30 min. Samples were dehydrated in graded ethanol series and propylene oxide, followed by a gradual embedding in Araldite 502 (Electron Microscopy Sciences, Fort Washington PA, USA). Sections were cut using Leica Ultracut UCT microtome (Leica Microsystems, Nussloch, Germany), contrasted by Uranyl acetate and lead citrate and observed in Jeol JEM-1230 TEM (JEOL LTD, Tokyo, Japan) at 80 kV. Electron micrographs were taken using TemCam-F214 (Tietz Video & Image Processing Systems (TVIPS) Gauting, Germany.

Images of dendritic networks (MAP2 staining) were manually converted into graphs (Figure 3a1,2) by implementing a graph deduction paradigm similar to the one used by Shefi et al. [17], where vertices were marked mainly at dendro-dendritic intersections, cell bodies, bifurcations, or growth cones (Figure 3a1,2). Dendritic segments connecting such vertices were considered as edges (Figure 3a2). Graphic reconstruction was performed via a custom MATLAB graphical user interface developed for this purpose (available for academic use at www.bgu.ac.il/~blinderp/public_html/software.html). The outcome of the graph abstraction process is a fully connected planar graph G (V, E), defined by the set V of its n vertices and E, the set of edges. Convenient representation and handling of such graphs involves the adjacency matrix, A, where each of its a_ij entries specifies the existence of a link (edge) between vertices i and j:(1)

The graph G is planar as any edge cross is considered a vertex and is made symmetric, (i.e. a_ij = a_ji), as no information regarding the direction of signal propagation along each edge (dendritic segment) is available from the real network.

The level of aggregation of the dendritic networks was estimated by implementing the relative neighborhood index (Ω_x) [18], [19]. For a given system with discrete elements in space, this analysis measures the aggregation (or dispersion) based on the change in annular density (d_x) over a specific spatial range of interest. Here, d_x is the number of DCs found in a ring divided by its area (a_x). Each ring is defined by x and x+Δ x, the inner and outer radii respectively. For each DC, a series of d_x values were computed by placing a set of concentric rings. In the current analysis, x ranged from 0 to 30 µm with Δx = 1 µm. Averaging this metric across the n DCs in a given network yields the global annular density D_x:

Finally, to compare between different networks, D_x is normalized by the absolute DC density d′; resulting in the relative neighborhood index, Ω_x = D_x/d′. The d′ metric consists of the number of DCs divided by the area of the network. Since the network is arbitrary defined by the extent of the field of view (see fig. 3a for example), the area of the network was measured in a polygon defined by connecting the outer most DCs. By definition [18], normalized values above 1 indicate a clustered spatial distribution, while values below 1 point toward spatial over-dispersion (i.e. more empty spaces).". It follows from this that –for a finite system–, if clustering is identified at a specific range, one should expect a complementary over-dispersion at a longer range.

Two complementary measurements of vertex synaptic weight were performed and implemented, depending on the specific analysis. While focusing on DCs per se (as previously described for the aggregation pattern), synaptic distribution was estimated as the total fluorescence measured in a disk of 5 µm radius. This approach is referred to here as the disk method. In turn, while considering the collective properties of groups of contacts (i.e. DCCs), vertex synaptic weight was computed by measuring total fluorescence in the area of a polygon (defined by the most extreme vertices of the DCC) intersected with an appropriate dendritic mask (Supplemental Material S1 section 2) and then dividing this value by the number of vertices in the DCC.

The usage of rats in this work was approved by the Ben Gurion University Committee for the Ethical Care and Use of Animal in experiments (Authorization number: IL-17-2-2005).